Question

the distance d you bike varies directly with the amount of time t you bike. suppose you bike 9.8 mi in 1.25 h. what is an equation that relates d and t? what is the graph of the equation?

write an equation that relates d and t. choose the correct answer below
a. ( d = 1.25t )
b. ( d = 9.8t )
c. ( d = 12.25t )
d. ( d = 7.84t )

graph the equation. choose the correct graph below
a. graph with d and t axes, line from origin
b. graph with d and t axes, shallow line from origin
c. graph with d and t axes, line from origin
d. graph with d and t axes, line from origin

Explanation:

Part 1: Equation relating \( d \) and \( t \)

Step 1: Recall direct variation formula

Direct variation means \( d = kt \), where \( k \) is the constant of proportionality.

Step 2: Calculate \( k \)

We know \( d = 9.8 \) mi when \( t = 1.25 \) h. Substitute into \( d = kt \):
\( 9.8 = k(1.25) \)

Step 3: Solve for \( k \)

Divide both sides by \( 1.25 \):
\( k = \frac{9.8}{1.25} = 7.84 \)

So the equation is \( d = 7.84t \), which corresponds to option D.

Part 2: Graph of the equation

Step 1: Analyze the equation \( d = 7.84t \)

This is a linear equation in the form \( y = mx \) (where \( d \) is \( y \), \( t \) is \( x \), and \( m = 7.84 \)). The slope \( 7.84 \) means for each unit increase in \( t \) (time), \( d \) (distance) increases by \( 7.84 \).

Step 2: Match the slope to the graph

• The graph should pass through the origin (since \( d = 0 \) when \( t = 0 \)).
• A slope of \( 7.84 \) is relatively steep (steeper than, e.g., a slope of 1). Among the options, the graph with a steep line through the origin (consistent with \( d = 7.84t \)) is the one that matches. Assuming the graphs are labeled with \( t \) on the x - axis and \( d \) on the y - axis, the correct graph will have a line with slope \( 7.84 \) (steep) starting at \( (0,0) \).

Final Answers

• Equation: \(\boldsymbol{d = 7.84t}\) (Option D)
• Graph: The graph with a steep line through the origin (consistent with the slope \( 7.84 \); specific graph identification depends on the visual options, but the key is the steep slope from \((0,0)\)).

Answer:

Part 1: Equation relating \( d \) and \( t \)

Step 1: Recall direct variation formula

Direct variation means \( d = kt \), where \( k \) is the constant of proportionality.

Step 2: Calculate \( k \)

We know \( d = 9.8 \) mi when \( t = 1.25 \) h. Substitute into \( d = kt \):
\( 9.8 = k(1.25) \)

Step 3: Solve for \( k \)

Divide both sides by \( 1.25 \):
\( k = \frac{9.8}{1.25} = 7.84 \)

So the equation is \( d = 7.84t \), which corresponds to option D.

Part 2: Graph of the equation

Step 1: Analyze the equation \( d = 7.84t \)

This is a linear equation in the form \( y = mx \) (where \( d \) is \( y \), \( t \) is \( x \), and \( m = 7.84 \)). The slope \( 7.84 \) means for each unit increase in \( t \) (time), \( d \) (distance) increases by \( 7.84 \).

Step 2: Match the slope to the graph

• The graph should pass through the origin (since \( d = 0 \) when \( t = 0 \)).
• A slope of \( 7.84 \) is relatively steep (steeper than, e.g., a slope of 1). Among the options, the graph with a steep line through the origin (consistent with \( d = 7.84t \)) is the one that matches. Assuming the graphs are labeled with \( t \) on the x - axis and \( d \) on the y - axis, the correct graph will have a line with slope \( 7.84 \) (steep) starting at \( (0,0) \).

Final Answers

• Equation: \(\boldsymbol{d = 7.84t}\) (Option D)
• Graph: The graph with a steep line through the origin (consistent with the slope \( 7.84 \); specific graph identification depends on the visual options, but the key is the steep slope from \((0,0)\)).